Manuscript Em01-D@2 · Em01-D-v2.tex · proposition · d3b49151

Proposition: Exact shifted cone equations

Characteristic coordinates and actual smoothing › Shifted equations for the actual relative variables · lines 1046–1103 · proof included or linked

Em01-D@2#prop:shifted, Em01-D@2#eq:shift1, Em01-D@2#eq:shift2, Em01-D@2#eq:scaledshift1, Em01-D@2#eq:scaledshift2, Em01-D@2#eq:scaledouter

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With $x_{1} = y - a$ , $x_{2} = y - b$ , the shifted equations are

(x_{1} - δ H_{1})^{2} (x_{2} - δ H_{2})^{2} = 4 (α - δ A_{1}) (v - δ B_{1}) - ∣ ξ - δ Z_{1} ∣^{2}, = 4 (d + δ A_{2}) (u - v + δ B_{2}) - ∣ ξ - δ Z_{2} ∣^{2} .

On the bounded rescaled core

u = δ U, v = δ V, ξ = δ Ξ,

their divided forms are

δ^{- 1} (x_{1} - δ H_{1})^{2} δ^{- 1} (x_{2} - δ H_{2})^{2} δ^{- 1} q_{R} = 4 (α - δ A_{1}) (V - B_{1}) - ∣Ξ - δ Z_{1} ∣^{2}, = 4 (d + δ A_{2}) (U - V + B_{2}) - ∣Ξ - δ Z_{2} ∣^{2}, = 4 β U .

The coordinate density in $(u, v, ξ)$ contributes $δ^{3}$ , with its full smooth factor $4 (β - δ U)^{2}$ .

All these algebraic formulas and their required derivatives are uniform on compact positive tangential parameter ranges and bounded rescaled cores.

Proof. In (eq:coordinates),

x_{2} = - d ℓ_{n} + (v - u) k_{n} + ξ^{A} E_{A} .

Subtracting $δ H_{2}$ and taking the square gives (eq:shift2); the first equation is analogous. Substitution gives the divided equations.

The two longitudinal normal variables contribute $δ^{2}$ , and the two transverse variables contribute $δ$ . Uniformity follows directly from the polynomial expressions and the bounded coordinate transformations.

Displayed formulas in this passage

Em01-D@2#eq:shift1 · line 1049
$(x_{1} - δ H_{1})^{2} (x_{2} - δ H_{2})^{2} = 4 (α - δ A_{1}) (v - δ B_{1}) - ∣ ξ - δ Z_{1} ∣^{2}, = 4 (d + δ A_{2}) (u - v + δ B_{2}) - ∣ ξ - δ Z_{2} ∣^{2} .$
Em01-D@2#eq:shift2 · line 1049
$(x_{1} - δ H_{1})^{2} (x_{2} - δ H_{2})^{2} = 4 (α - δ A_{1}) (v - δ B_{1}) - ∣ ξ - δ Z_{1} ∣^{2}, = 4 (d + δ A_{2}) (u - v + δ B_{2}) - ∣ ξ - δ Z_{2} ∣^{2} .$
Em01-D@2#eq:scaledshift1 · line 1067
$δ^{- 1} (x_{1} - δ H_{1})^{2} δ^{- 1} (x_{2} - δ H_{2})^{2} δ^{- 1} q_{R} = 4 (α - δ A_{1}) (V - B_{1}) - ∣Ξ - δ Z_{1} ∣^{2}, = 4 (d + δ A_{2}) (U - V + B_{2}) - ∣Ξ - δ Z_{2} ∣^{2}, = 4 β U .$
Em01-D@2#eq:scaledshift2 · line 1067
$δ^{- 1} (x_{1} - δ H_{1})^{2} δ^{- 1} (x_{2} - δ H_{2})^{2} δ^{- 1} q_{R} = 4 (α - δ A_{1}) (V - B_{1}) - ∣Ξ - δ Z_{1} ∣^{2}, = 4 (d + δ A_{2}) (U - V + B_{2}) - ∣Ξ - δ Z_{2} ∣^{2}, = 4 β U .$
Em01-D@2#eq:scaledouter · line 1067
$δ^{- 1} (x_{1} - δ H_{1})^{2} δ^{- 1} (x_{2} - δ H_{2})^{2} δ^{- 1} q_{R} = 4 (α - δ A_{1}) (V - B_{1}) - ∣Ξ - δ Z_{1} ∣^{2}, = 4 (d + δ A_{2}) (U - V + B_{2}) - ∣Ξ - δ Z_{2} ∣^{2}, = 4 β U .$
unlabeled · line 1062
$u = δ U, v = δ V, ξ = δ Ξ,$
unlabeled · line 1092
$x_{2} = - d ℓ_{n} + (v - u) k_{n} + ξ^{A} E_{A} .$

Proposition: Exact shifted cone equations

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